Concept Byte: The Ambiguous Case

For Use With Lesson 14-4

The triangles below have one pair of congruent angles and two pairs of congruent sides. But the triangles are not congruent. Notice that the congruent angles are not included by the congruent sides.

When you know the measures of two sides of a triangle and one of the opposite angles, there may be two triangles with those measurements. You can use the Law of Sines to find the other measures for both triangles.

Example

In each Δ ABC cap delta below, m ∠ A = m angle eh equals 35°, a = equals 11, and b = b equals 15. Find m ∠ B m angle b .

Image Long Description

sin A a = sin B b Law of Sines sin 35 ° 11 = sin B 15 Substitute . sin B = 15 sin 35 ° 11 Solve for sin B . m ∠ B = sin − 1 ( 15 sin 35 ° 11 ) ≈ 51 ° Solve for  m ∠ B . Use a calculator . table with 4 rows and 3 columns , row1 column 1 , fraction sine eh , over eh end fraction , column 2 equals . fraction sine b , over b end fraction , column 3 cap lawofcap sines , row2 column 1 , fraction sine , 35 degrees , over 11 end fraction , column 2 equals . fraction sine b , over 15 end fraction , column 3 cap substitute . . , row3 column 1 , sine b , column 2 equals . fraction 15 sine , 35 degrees , over 11 end fraction , column 3 cap solvefor sine b . , row4 column 1 , m angle b , column 2 equals . sine super negative 1 end super . open . fraction 15 sine , 35 degrees , over 11 end fraction . close . almost equal to 51 degrees , column 3 cap solvefor . m angle b . . cap useacalculator . . , end table

The sine function is also positive in Quadrant II. So another value of m ∠ B m angle b is about 180 ° − 51 ° = 129 ° . 180 degrees negative 51 degrees equals 129 degrees .

Because there are two possible angle measures for ∠ B , angle b comma there are two triangles that satisfy the given conditions. In one triangle the angle measures are about 35 ° , 51 ° , 35 degrees comma . 51 degrees comma and 94 ° . 94 degrees . In the other, the angle measures are about 35 ° , 129 ° , 35 degrees comma . 129 degrees comma and 16 ° . 16 degrees .

Exercises

In each Δ ABC , cap delta find the measures for ∠ B angle b and ∠ C angle c that satisfy the given conditions. Draw diagrams to help you decide whether two triangles are possible.

1. m ∠ A = 62 ° , a = 30 , m angle eh equals , 62 degrees comma , eh equals 30 comma and b = 32 b equals 32
2. m ∠ A = 16 ° , a = 12 , m angle eh equals , 16 degrees comma , eh equals 12 comma and b = 37.5 b equals , 37.5
3. m ∠ A = 48 ° , a = 93 , m angle eh equals , 48 degrees comma , eh equals 93 comma and b = 125 b equals 125
4. m ∠ A = 112 ° , a = 16.5 m angle eh equals , 112 degrees comma , eh equals , 16.5 , and b = 5.4 b equals 5.4
5. m ∠ A = 23.68 m angle eh equals , 23.68 , a = 9.8 eh equals 9.8 , and b = 17 b equals 17
6. m ∠ A = 155 ° , a = 12.5 m angle eh equals , 155 degrees comma , eh equals , 12.5 , and b = 8.4 b equals 8.4
7. Multiple Choice You can construct a triangle with compass and straightedge when given three parts of the triangle (except for three angles). Which of the following given sets could result in the ambiguous case?
   1. Given: three sides
   2. Given: two sides and an included angle
   3. Given: two sides and a non-included angle
   4. Given: two angles and a non-included side

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